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Markov-Modulated Brownian Motion with Two Reflecting Barriers

Part of:Special processesMarkov processes

Published online by Cambridge University Press:  14 July 2016

Jevgenijs Ivanovs
Jevgenijs Ivanovs*
Affiliation:
EURANDOM, Eindhoven University of Technology and University of Amsterdam
*
∗ Postal address: EURANDOM, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: ivanovs@eurandom.tue.nl
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Abstract

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We consider a Markov-modulated Brownian motion reflected to stay in a strip [0, B]. The stationary distribution of this process is known to have a simple form under some assumptions. We provide a short probabilistic argument leading to this result and explain its simplicity. Moreover, this argument allows for generalizations including the distribution of the reflected process at an independent, exponentially distributed epoch. Our second contribution concerns transient behavior of the model. We identify the joint law of the processes defining the model at inverse local times.

Keywords

Markov additive processfluid modelfinite buffertwo-sided reflectioninverse local timeoverflow process

MSC classification

Secondary:60J55: Local time and additive functionals60K25: Queueing theory60K37: Processes in random environments

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010, pp. 1034 - 1047
Copyright
Copyright © Applied Probability Trust 2010 

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